LATTCCHEMISTRY 101

Hydrogen Spectrum and Bohr’s Theory

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Energies of electrons in atoms are quantized; this is evidenced by the emission (line) spectra of atoms.

Atoms of an element emit a unique set of specific wavelengths (spectral lines) that are the characteristic of that

element. Line spectra of elements are the elemental equivalent of fingerprints, because each element has a line

spectrum different from all of the other elements.

The energy levels and the spectrum of the hydrogen atom can be accounted for by using Bohr’s equation:

n = 1, 2, 3, – – – – – – -, ,

En = –RH / n2,

(1)

where RH = 2.179910–18 J is the Rydberg constant, and n is an integer, called the principal quantum number.

The spectral line of an atomic spectrum is produced when an electron absorbs energy and is excited to a

higher energy level. The excited state is not stable, thus the electron quickly returns to a lower energy state and

emits energy in the form of a photon, which is shown as a fine spectral line recorded on a photographic plate

(see Fig. 1). If one knows the energy of each level, the energy of transition from a higher level, Ei, to a lower

one, Ef, can be computed.

E = Ef – Ei = –RH (1/nf2 – 1/ni2)

(2)

where nf and ni are principal quantum numbers of the final and initial energy levels, respectively.

On the other hand, the energy of a photon is also proportional to the frequency, , which in turn is

inversely proportional to the wavelength, .

E = h = hc / ,

(3)

where h = 6.625610–34 J.s is the Planck’s constant and

c = 2.9979108 m/s is the speed of light

Rearranging Eq. (3), we obtain the wavelength of a hydrogen spectral line.

= hc / E ,

(4)

The value of hc can be pre-computed to give

= hc / E = (6.625610–34 J.s) (2.9979108 m/s) / (E)

= (1.986310–25 / E) m.J

= (1.986310–16 / E) nm.J

(5)

The computational steps of obtaining the wavelength is:

1. Calculate the energy levels, En, using Eq. (1)

2. Calculate the energy difference, E, of a given transition by using Eq. (2)

3. Substitute the E value into Eq. (5) to obtain the wavelength in the unit of nm.

PROCEDURE

1. Hydrogen Energy Levels

Calculate the first six energy levels of the hydrogen atom using Eq. (1) and insert the results in the right-hand side of Fig. 2.

Table 1. Energy levels of the hydrogen atom

Principal quantum

Energy, En (J)

number, n

1

2

3

4

5

6

2. Wavelengths of the Hydrogen Spectrum

Calculate the wavelengths of the specified transitions in Table 2 using Equations (2) and (5).

n_initial –> n_final

Transitions

61

Table 2. Wavelengths of the hydrogen spectral lines.

Energy difference, E (J) (use equation 2)

Wavelength (nm) (equation 5)

51

41

31

21

62

52

42

32

63

53

43

64

54

65

3. Identification of Transitions

3a. Identify the wavelengths of the spectral lines of hydrogen from Figure 1 and insert those values in column 1 of Table 3.

3b. Identify the corresponding transitions (wavelengths) of these spectral lines using Table 2 values.

i. Put the matched wavelengths from Table 2 in the second column of Table 3.

ii. Put the corresponding transitions from Table 2 in the third column of Table 3.

3c. Draw arrows for the transitions shown in table 3, from initial state (higher n value) to final state (lower n value) with

the arrow pointed down in Figure 2, and also fill in the blanks for the energy levels and corresponding energy values.

Estimate the ‘experimental’ wavelengths in Figure1 and put in Column 1. Get calculated wavelengths from Table 2 and match them with Column1 wavelengths.

Place corresponding transitions (example 4–> 2) from Table 2 into Column 3. Now we know which experimental wavelength corresponds to which transition.

Table 3. Hydrogen’s spectral lines and the calculated spectral lines and their corresponding transitions using the Bohr equation.

Wavelength (nm) of hydrogen’s

Calculated wavelength (nm)

Transitions

spectral lines (from Fig. 1)

(from Table 2)

(from Table 2)

Use Table 1 data to fill in the below.

0

E n ( J)

–

n=

n = En =

n = En =

n = En =

n=

En =

n=

En =

n=

En =

–

–

–

Discussion Questions

1. How well does the Bohr equation predict the line spectrum of hydrogen?

2. Which transition in table 3 emits the highest energy photon?

3. Which transition in table 3 emits the lowest energy photon?

4. Which transition in table 3 emits the highest energy photon in the visible region?

5. Which transition in table 3 emits the lowest energy photon in the visible region?

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